Using the Identity and Inverse Properties of Addition and Subtraction

Learning Outcomes

Identify the identity properties of multiplication and addition
Use the inverse property of addition and multiplication to simplify expressions

Recognize the Identity Properties of Addition and Multiplication

What happens when we add zero to any number? Adding zero doesn’t change the value. For this reason, we call $0$ the additive identity.

For example,

$\begin{array}{ccccc}\hfill 13+0\hfill & & \hfill -14+0\hfill & & \hfill 0+\left(-3x\right)\hfill \\ \hfill 13\hfill & & \hfill -14\hfill & & \hfill -3x\hfill \end{array}$

What happens when you multiply any number by one? Multiplying by one doesn’t change the value. So we call $1$ the multiplicative identity.

For example,

$\begin{array}{ccccc}\hfill 43\cdot 1\hfill & & \hfill -27\cdot 1\hfill & & \hfill 1\cdot \frac{6y}{5}\hfill \\ \hfill 43\hfill & & \hfill -27\hfill & & \hfill \frac{6y}{5}\hfill \end{array}$

Identity Properties

The Identity Property of Addition: for any real number $a$ ,

$\begin{array}{}\\ \hfill a+0=a(0)+a=a\hfill \\ \hfill \text{0 is called the}\mathbf{\text{ additive identity}}\hfill \end{array}$

The Identity Property of Multiplication: for any real number $a$

$\begin{array}{c}\hfill a\cdot 1=a(1)\cdot a=a\hfill \\ \hfill \text{1 is called the}\mathbf{\text{ multiplicative identity}}\hfill \end{array}$

example

Identify whether each equation demonstrates the identity property of addition or multiplication.

1. $7+0=7$
2. $-16\left(1\right)=-16$

Solution:

1.
$7+0=7$
We are adding 0.	We are using the identity property of addition.

2.
$-16\left(1\right)=-16$
We are multiplying by 1.	We are using the identity property of multiplication.

try it

Use the Inverse Properties of Addition and Multiplication

What number added to 5 gives the additive identity, 0?
$5 + =0$	We know $5+(\color {red}{--5})=0$
What number added to −6 gives the additive identity, 0?
$-6 + =0$	We know $--6+\color {red}{6}=0$

Notice that in each case, the missing number was the opposite of the number.

We call $-a$ the additive inverse of $a$ . The opposite of a number is its additive inverse. A number and its opposite add to $0$ , which is the additive identity.

What number multiplied by $\Large\frac{2}{3}$ gives the multiplicative identity, $1?$ In other words, two-thirds times what results in $1?$

\frac{2}{3} \cdot = 1

$\Large\frac{2}{3}\normalsize\cdot =1$

We know

\frac{2}{3} \cdot \frac{3}{2} = 1

$\Large\frac{2}{3}\normalsize\cdot\color{red}{\Large\frac{3}{2}}\normalsize=1$

What number multiplied by $2$ gives the multiplicative identity, $1?$ In other words two times what results in $1?$

2 \cdot = 1

$2\cdot =1$

We know

2 \cdot \frac{1}{2} = 1

$2\cdot\color{red}{\Large\frac{1}{2}}\normalsize=1$

Notice that in each case, the missing number was the reciprocal of the number.

We call $\Large\frac{1}{a}$ the multiplicative inverse of $a\left(a\ne 0\right)\text{.}$ The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to $1$ , which is the multiplicative identity.

We’ll formally state the Inverse Properties here:

Inverse Properties

Inverse Property of Addition for any real number $a$ ,

$\begin{array}{}\\ \hfill a+\left(-a\right)=0\hfill \\ \hfill -a\text{ is the}\mathbf{\text{ additive inverse }}\text{of }a.\hfill \end{array}$

Inverse Property of Multiplication for any real number $a\ne 0$ ,

$\begin{array}{}\\ \\ \hfill a\cdot \frac{1}{a}=1\hfill \\ \hfill \frac{1}{a}\text{is the}\mathbf{\text{ multiplicative inverse }}\text{of }a.\hfill \end{array}$

example

Find the additive inverse of each expression:
1. $13$
2. $-\Large\frac{5}{8}$
3. $0.6$

Show Solution

Solution:
To find the additive inverse, we find the opposite.
1. The additive inverse of $13$ is its opposite, $-13$
2. The additive inverse of $-\Large\frac{5}{8}$ is its opposite, $\Large\frac{5}{8}$
3. The additive inverse of $0.6$ is its opposite, $-0.6$

try it

example

Find the multiplicative inverse:
1. $9$
2. $-\Large\frac{1}{9}$
3. $0.9$

Show Solution

Solution:
To find the multiplicative inverse, we find the reciprocal.
1. The multiplicative inverse of $9$ is its reciprocal, $\Large\frac{1}{9}$
2. The multiplicative inverse of $-\Large\frac{1}{9}$ is its reciprocal, $-9$
3. To find the multiplicative inverse of $0.9$ , we first convert $0.9$ to a fraction, $\Large\frac{9}{10}$ . Then we find the reciprocal, $\Large\frac{10}{9}$

Module 8: Real Numbers